Nature-Inspired Intelligent Computing: A Comprehensive Survey

Nature, with its numerous surprising rules, serves as a rich source of creativity for the development of artificial intelligence, inspiring researchers to create several nature-inspired intelligent computing paradigms based on natural mechanisms. Over the past decades, these paradigms have revealed effective and flexible solutions to practical and complex problems. This paper summarizes the natural mechanisms of diverse advanced nature-inspired intelligent computing paradigms, which provide valuable lessons for building general-purpose machines capable of adapting to the environment autonomously. According to the natural mechanisms, we classify nature-inspired intelligent computing paradigms into 4 types: evolutionary-based, biological-based, social-cultural-based, and science-based. Moreover, this paper also illustrates the interrelationship between these paradigms and natural mechanisms, as well as their real-world applications, offering a comprehensive algorithmic foundation for mitigating unreasonable metaphors. Finally, based on the detailed analysis of natural mechanisms, the challenges of current nature-inspired paradigms and promising future research directions are presented.


Introduction
For centuries, philosophers and scientists have attempted to create increasingly complex artificial systems by observing, understanding, explaining, and imitating many interesting phenomena in nature.Over hundreds of millions of years, a vast number of species have been born in the universe, the underlying complexity of which is far beyond human comprehension.These species often confront each other with constraints and form part of natural systems.The evolution of complexity in nature tends to follow a low-to-high distribution.So far, natural phenomena, known or unknown, have become a source of creativity in various new disciplines and technological fields [1][2][3][4][5].
With the invention of computers, researchers have consistently looked to nature for inspiration to build general-purpose machines that rivaled or even surpassed humans.Due to its parallelism, derivative-free, and easy scalability, nature-inspired intelligent computing paradigms have proven to be very successful in solving practically complex NP-hard problems and black-box optimization problems such as aerodynamic optimization with an expensive evaluation process [2].A series of studies [3][4][5] have shown that these paradigms are powerful methods for cognition and optimization, such as optimizing complex dynamical systems and building cognitive models.
Nature-inspired intelligent computing paradigms model various phenomena in the natural world to solve various practical problems faced by humans, effectively building a "bridge" between natural disciplines and artificial machines.For example, in nature, the evolved human brain allows humans to participate in complex social collaborations.Similarly, it is a hot application for the nature-inspired intelligent computing paradigm to use evolutionary algorithms (EAs) to optimize braininspired artificial neural networks (ANNs) so that disorganized neural networks can evolve into general-purpose neural networks [6][7][8].As shown in Fig. 1, ANNs can be employed for cognition, learning, and reasoning by evolving the architecture A and parameters W of the ANNs.
In general, nature-inspired intelligent computing paradigms have characteristics that nature possesses, such as randomness, nonlinearity, ergodicity, self-organization, adaptability, diversity, stability, and high parallelism.Due to the improvement in computing and information processing capabilities of computers over the past decade, nature-inspired intelligent computing paradigms are employed to solve various complex problems in the real world.For instance, material design can be optimized using genetic algorithms (GAs), while material design can be effectively addressed through GA methods [9].River flow forecasting has been successfully performed using EAs and particle swarm optimization (PSO) techniques [10].Additionally, quantum-inspired gray wolf optimization (GWO) has been applied for breast cancer diagnosis [11], and intelligent inspection robots for obstacle avoidance have been constructed using improved PSO [12].Although providing systematic solutions for highly complex problems, nature-inspired intelligent computing paradigms are dwarfed by the diversity, stability, and adaptability of nature.
Although existing reviews [13][14][15][16][17][18][19][20][21][22][23][24][25][26] have offered valuable insights, a comprehensive survey that systematically encompasses all categories within the nature-inspired intelligent computing paradigm is still lacking.Furthermore, the connections between natural mechanisms and these paradigms, as well as the deep mathematical or physical principles underlying them, are not fully explored.This paper aims to fill the gap by presenting a wide-ranging exploration of nature-inspired intelligent computing research, including evolution-based, biological-based [encompassing social behaviors and immune systems (ISs)], culture-based, and science-based methods as shown in Fig. 2. We delve into various nature-inspired intelligent computing paradigms from a natural mechanism perspective, focusing on the origins of the algorithms.This perspective reveals the nuances that distinguish natural mechanisms from natureinspired intelligent computing paradigms, uncovers commonalities, and encapsulates key design principles, updating procedures, and typical applications.In addition, this paper summarizes potential future challenges and discusses prospective advancements in several application areas.The main contributions of this paper are as follows: 1. Comprehensive overview: We provide a comprehensive overview of the natural mechanisms of the nature-inspired intelligent computing paradigms and divide them into 4 major categories: evolution-based, biological-based, social-culturalbased, and science-based.This holistic approach provides a broader understanding of the field.
2. Characteristics and real-world applications: This paper investigates the commonalities as well as classic real-world application examples of nature-inspired intelligent computing paradigms.These reviews reveal connections between natural mechanisms and nature-inspired intelligent computing paradigms.
3. Connections between natural mechanisms and paradigms: We delve into the connections between natural mechanisms and the corresponding computational paradigms, uncovering the fundamental principles and design methodologies of algorithms inspired by nature.This perspective highlights the nuances that distinguish natural mechanisms from nature-inspired computing paradigms and encapsulates key design principles, updating procedures, and typical applications.The organization of this paper is as follows.In the "Related Work" section, we review related work on nature-inspired intelligent computing paradigms.The "Evolution-Based Paradigms" section presents the natural mechanisms of evolution-based paradigms, while the "Biological-Based Paradigms" section discusses those of biological-based paradigms.The fundamental natural mechanisms of social-cultural-based paradigms are detailed in the "Social-Cultural-Based Paradigms" section, and the natural mechanisms of science-based paradigms are outlined in the "Science-Based Paradigms" section.The "Challenges and Potential Future Research Directions" section points out the challenges of the nature-inspired intelligent computing paradigms and potential future research directions.Finally, the "Summary and Outlook" section concludes this paper.

Related Work
Many surveys and reviews have been published to highlight the pivotal role of nature-inspired intelligent computing paradigms, each of which provides valuable insights and a comprehensive overview of the field.These scholarly contributions enrich our understanding of the fundamental concepts, recent advancements, and prospective future trajectories in these paradigms.
In the field of nature-inspired intelligent computation, researchers have delved into a myriad of paradigms, with their own unique perspectives.For example, Kumar and Singh [15] and Sachan and Kushwaha [16] offered comprehensive reviews covering evolution, swarm intelligence, biology, and scientific approaches, and scientific viewpoints.Tang et al. [17] delved specifically into swarm intelligence algorithms, whereas Chelly Dagdia et al. [14] focused on the interplay between biology and

Genotype-phenotype Mapping
Fig. 1.The nature-inspired intelligent computing paradigm provides an approach to evolving the architecture and parameters of an ANN, allowing it to dynamically interact with its environment for cognition, learning, and reasoning.
computer science.Many works [13,19] emphasized the connection between evolutionary methodologies and evolutionary computation.Torres-Treviño [18] explored biological and viral standpoints.Omidvar et al. [4,27] summarized metaheuristics for large-scale problems.Recently, Kudela [28] analyzed the classic evolutionary paradigms on benchmark functions and gave several suggestions for improvement.Gharehchopogh [29] provided a comprehensive summary of quantum-inspired metaheuristics and their applications in science and engineering.For paradigm descriptions, the existing surveys focused on additional procedural steps, variations of algorithms, and their diverse applications [15][16][17][18].Alternatively, some work [13,14,19] opted for a more synthesized overview of specific algorithm classes.These explorations highlighted the powerful applicability of nature-inspired intelligent computing paradigms, illustrating its expansive scope and dynamic evolution.
This survey builds upon the valuable insights from existing reviews by offering a more comprehensive and integrative perspective on nature-inspired intelligent computing paradigms.It distinguishes itself by systematically encompassing all major categories of nature-inspired computing paradigms, including evolution-based, biological-based, social-cultural-based, and science-based methods.The work delves into the deep connections between natural mechanisms and their corresponding computational paradigms, uncovering fundamental principles and design methodologies.Beyond surface descriptions, it explores the deep mathematical principles and classic real-world application examples, providing a deeper understanding of their origins, operational mechanisms, and practical applications.Additionally, the survey offers a detailed discussion of current challenges and future research directions, presenting a comprehensive roadmap for future exploration and potential advancements in the field.By focusing on these aspects, the survey aims to provide a holistic and integrative understanding of nature-inspired intelligent computing paradigms, bridging gaps in the existing literature and paving the way for future research.

Evolution-Based Paradigms
Evolution provides a creative source of power for complex and delicate problems.Inspired by the origin of species proposed by Darwin [38], evolution-based paradigms have become one of the best-known concepts in the field of nature-inspired intelligent computing paradigms [39].In a sense, evolution is independent of various physical media and occurs in the development of anything.Like biological evolution, evolution-based paradigms exhibit surprising experimental results on complex problems [40,41].In this section, the mechanisms of natural evolution and artificial evolution paradigms, as well as the gap between them are systematically reviewed.

The natural mechanism of evolution-based paradigms
Since the Big Bang, all kinds of primitive life have gradually reproduced and evolved into more complex life forms on land, sea, and air.It can be observed that evolutionary adaptation leads to a diversity of organisms.Over time, species make adaptive adjustments to environmental changes.Differences in genes cause individuals of the same species to exhibit different variation characteristics.Individuals who successfully adapt to their environment are more likely to survive and reproduce so that the good genes are passed on to their offspring.Those individuals who cannot adapt to the environment are eliminated.Inspired by this evolutionary process, the powerful creativity of evolution is used to solve complex problems in many fields such as natural cognition, scientific exploration, and social development.For nearly a hundred years, several underlying mechanisms of inheritance have been revealed by quantitatively exploring the spread of variant alleles, leading to a unified interpretation of biodiversity on Earth [13,14].Key drivers of evolution include genotype-to-phenotype mapping, Genotype-to-phenotype mapping is fundamental to the evolutionary process of micronized analysis.Gene refers to the basic material unit that carries genetic information.The properties of an organism are inherited through genes or other mechanisms.Humans contain about 20,000 to 25,000 genes.The interaction of genotype and environment can produce a set of observable traits, called phenotypes, such as biological structure or behavior.By exploring the relationship between genotype and phenotype, the functions of many genes have been revealed [42].Mapping between genotypes and phenotypes is a large-scale complex nonlinear problem, which is difficult due to too little data on genotypes, insufficient description of phenotypes, and the underlying complexity of regulating cellular functional networks [43].In general, genotype and phenotype are not in a one-to-one mapping and they are both regulated by gene regulatory networks and phenotypic genetic traits [44].Most traits are controlled by multiple interacting genes [45].For example, the combinatorial complexity of potential epistasis linking in proteins makes relating genotype and phenotype difficult [46].In addition, the organism itself also possesses phenotypic plasticity [47], that is, when the genotype remains unchanged, the phenotype may also change and be passed on to the next generation.
Mutation, genetic drift, recombination, and gene flow are the creative sources of evolution, providing the raw material for natural selection.In the process of reproduction, mutation is the change of genetic material at a certain frequency, such as point mutation caused by a single base [48] and deletion, insertion, and rearrangement of multiple bases [49].The triggers include replication errors during cell division [49], chemicals (nitrous acid [50], aflatoxin [51], etc.), radiation (x-rays [52], ultraviolet light [53], etc.), or the effects of viruses [54].Mutation frequencies vary widely across species, strongly influenced by the environment and the genetic information of the organism [55].Exploring mutation frequencies can prioritize genes and pathways in a manner that increases public health benefits [56].
Mutations are universal, random, rare, reversible, nondirectional, independent, and recurring.The effects of mutations on an individual may be beneficial [57], detrimental [58], or both [59].Beneficial mutations increase the fitness of an organism.In recent studies, gene editing techniques could allow the introduction of beneficial mutations to improve genetic diseases [57], which further reveals the mechanism of complex gene expression, while harmful mutations can greatly reduce the fitness of organisms, for example, mutation-induced genes may lead to various diseases [58].Mutations that are neither beneficial nor harmful to the individual are called neutral mutations.In the absence of natural selection, neutral mutations still occur with a certain frequency and then accumulate gradually between species.Although neutral mutations have no direct effect on phenotype, studies [60] have shown that a robust neutral mutation may be more likely than a vulnerable mutation to spread in a large and diverse population and the neutral mutation sequence follows the "random walk with maximum entropy".
Genetic drift refers to the random variation of allele frequencies in a population between each generation.Even in the absence of selection, genetic drift causes different populations of the same genetic structure to evolve into new populations with different sets of alleles [61].Recombination is the exchange of genetic material between organisms, which allows offspring to own different characteristics from the parent.In general, prokaryotes can directly exchange genes with each other through conjugation, while eukaryotes perform gene recombination by exchanging chromosomes during meiosis and mitosis.The phenomenon of recombination is used for DNA repair to maintain genome integrity [62].Gene flow is the exchange of genes that occurs between populations or species, which is a new source of variation in populations or species.The transfer of genetic material between species leads to the formation of new species.For example, the eukaryotic genome is a fusion of different prokaryotic genomes [63].
Natural selection is the core force of evolution.Through natural selection, those traits with higher survivability become more prevalent in populations.Fitness is employed to measure an organism's ability to survive and reproduce.If an organism is more likely to survive and reproduce quickly, it has a higher fitness, and vice versa.Natural selection can act on different hierarchies, such as genes, cells, individuals, biota, and species, separately or simultaneously [64].Furthermore, sexual selection is a special case of natural selection.Traits that have evolved through sexual selection are particularly prominent in the males of some animal species.As shown in Fig. 3, recent studies [65] have shown that it is even possible to increase female fitness by reversing the exaggeration of male sexual selection traits.Fig. 3. Male and female phenotypes result from male-limited predation and male sexual selection.Sexual selection leads to enlarged mandibles in males, and male-restricted predation leads to larger abdomens in both males and females (image from [65]).

Evolution-inspired paradigms
The evolution-inspired paradigm is one of the oldest and most well-established branches of natural computation, originally developed for the study of adaptive systems rather than function optimization.Evolution-inspired paradigms encompass a wider variety of systems than function optimizers.The classic evolution-inspired paradigm encompasses 4 domains: GA, evolutionary programming (EP), evolutionary strategies (ES), and genetic programming (GP).In fact, hybrid paradigms of these 4 methods are becoming more and more popular.Their joint development leads to the prosperity of the evolutioninspired paradigm, which has been widely used in various engineering optimization problems [66].
Figure 4 presents a general framework based on the evolution-inspired paradigm.Inspired by genetic evolution, the evolution-based paradigm includes initialization, variation, evaluation, and selection.The whole process acts on many individuals or chromosomes.Each individual or chromosome represents a potential solution to a problem and is encoded according to the problem properties.Through reproductive operations (crossover and mutation) between individuals, those new traits or genes are constantly emerging in the population.There are some properties of the parent and new properties in the new individuals.Fitness is employed to measure the individual's performance on the problem to be solved.Individuals with high fitness gradually dominate the population, while individuals with lower fitness values are eliminated.Over a certain number of generations, individuals evolve to maximize their performance on the problem.Inspired by the concept of evolution, a series of different evolution-based paradigms have been proposed.These methods have different ways of expressing information, which are presented as follows.
Genetic algorithm.GA [67], a classical paradigm inspired by evolution, is originally employed to describe the adaptive process of natural systems [68], and later widely applied in learning and optimization.GAs maintain a population approaching the optimal solution by modeling the selection, crossover, and mutation mechanisms in natural evolution.
Classical GAs employ binary strings to represent an individual.Subsequently, the representation of individuals is also extended to other types, such as integer encoding, real encoding, and sequential encoding.Unlike the complex genotype-tophenotype mapping in natural evolution, individual representations in GAs tend to be one-to-one.
For the selection process, the algorithm designs uncertain selection operators such as roulette and binary tournaments.The update process of the population is often divided into 2 strategies: generational replacement and steady-state replacement [14].Generation replacement is the complete replacement of the parent with all the resulting children.Steady-state replacement replaces only a portion of the parent with the child.In survival selection, selection pressure refers to the degree to which excellent individuals are favored.The greater the selection pressure, the faster the population converges and the search space may not be fully explored.As the core driving force of evolution, suitable selective pressure ensures the spread of excellent genes and the elimination of inferior genes, pushing populations toward better solutions [69].
Crossover-dominant and mutation-secondary strategies are employed in GAs.Crossover operators include singlepoint, multi-point, and uniform crossover.Mutation operators include single-point, multi-point, and uniform mutation.In GAs, various problem-specific genetic operators can be designed to improve the performance of the algorithm.For example, in numerical optimization, simulated binary crossover [70] and polynomial mutation [71] are proposed to explore the search space of real numbers.
In addition, unlike the natural evolution theory, some local search strategies are introduced into the GAs to guide the population to speed up the evolution according to the specific prior knowledge of the solving problem, such as the memetic algorithm [72].Recently, many works utilizing neural networks to learn and distill GAs have been proposed to improve performance and generalization [73][74][75].Evolutionary strategies.ESs were proposed by Rechenberg [76] for solving continuous optimization problems.The individuals in its population are represented by genetic material and control parameters.The control parameters are usually a set of random vectors that are normally distributed.This parameter describes the specific behavior of the individual.In most ESs, the selection process is deterministic based on fitness ranking.In the process of evolution, control parameters and genetic material evolve at the same time.
The covariance matrix adaptation evolution strategy (CMA-ES), one of the most theoretically complete ESs [77], is widely used in various nonlinear or nonconvex continuous optimization problems.In CMA-ES, new individuals are generated from a multivariate normal distribution.Crossover is a process to select a new mean for the distribution, and variation is equivalent to adding a perturbation with zero means.The relationship between variables in distribution is represented by a covariance matrix.The adaptation of the covariance matrix is a method of automatically updating the distribution.On the quadratic function, this adaptive covariance matrix approximates the inverse of the Hessian matrix in the quasi-Newton method.The latest research [78] shows that CMA-ES is an information geometry optimization algorithm (IGOA) whose parameter distribution is a multivariate normal distribution.IGOA transforms any family of smooth parameter probability distributions over the search space into a black-box optimization algorithm in continuous time.The resulting IGOA is a flow of ordinary differential equations that guide the adaptive transformation of the objective function for natural gradient ascent.This idea provides a solid theoretical foundation for the ESs.Recently, Riemannian natural gradients and Hessian approximation have been further extensively studied to improve the theoretical performance of ESs [5,[78][79][80][81].
Evolutionary programming.EP [82] employs evolution as a learning process to generate a wide range of AI machines.It emphasizes changes in group behavior in natural evolution rather than genotype, so mutation is the only reproductive operator.Like ESs, probability distributions are used to guide the mutation process and distribution parameters are adaptively updated in EP.In EP, the selection mechanism of the parent is deterministic, while the replacement process is uncertain, such as Boltzmann selection and proportional selection.EP currently has no fixed structure, and the boundaries with several other paradigms are increasingly blurred.
Genetic programming.GP [83], a technique for automating the generation and selection of computer programs to accomplish user-defined tasks, is regarded as an important extension of GAs.A nonlinear structure like a tree is used to represent programs in the original GP.Similar to GAs, a group of randomly generated programs gradually evolve into programs adapted to specific tasks by operations such as crossover, mutation, and selection.GPs based on different data structures are also designed to generate modern computer programs, such as Python code generation [84].With the rapid development of deep learning, automatic deep learning has become an important application fileds of GP [85][86][87].The performance of deep learning is heavily influenced by its parameters as well as its architecture.Automatic deep learning is a constrained optimization problem with multi-level multiobjective high-dimensional characteristics, and its inherent nonconvex and black-box properties make GP one of the potential solutions.Recently, large language models (LLMs) have demonstrated extraordinary sequence learning capabilities [88].Accelerating GP using LLMs has been used to solve various complex mathematical reasoning and engineering optimization problems [89][90][91][92].Due to the flexibility and simplicity of language representation, LLM-assisted GP is expected to address more complex scenarios in open environments.

Biological-Based Paradigms
Biological-based paradigms stem from the study of "emergent" phenomena in nature.Such paradigms usually consist of a group of simple individuals that communicate with each other and their environment [93].Due to the highly self-organizing, parallel, stochastic, and simple implementation properties, biological-based paradigms have received broad attention among scholars and applied to various optimization problems.As shown in Fig. 5, this section starts from social biomechanics and extends a family of related paradigms based on social behaviors as well as the artificial immune system (AIS).

Social animal and swarm behavior
The movement of species aggregation is called swarm behavior [94].All these swarm behaviors are observed in nature, with ants cooperating with each other to forage and build complex burrow passages, birds avoiding collisions in flight to achieve coordination, and bees collecting nectar from food sources (flowers) with great efficiency.The "symphony-like" way in which swarms cooperate has always fascinated researchers.Every swarm consists of a number of simple entities with local interactions (including with the environment).The emergence of complex or mesomorphic behaviors and the ability of swarms to achieve significant results are the product of a combination of simple or microscopic behaviors.Social learning has been found in a variety of social animals or plants, where species learn by observing the behavior and experiences of other individuals as well as by the transmission of information between individuals within intraspecific species [95].Some social animals have evolved the flexibility and intelligence to deceive or benefit from other animals, even to predict others' behaviors [96].Research [97] showed that wild vervet monkeys are capable of learning what to eat from the more experienced individuals in their social group.Social animals coordinate their behavior in groups, and their nervous systems are likely to do the same [98].The theoretical foundations of biological research inspired researchers with numerous insights.

Social behavior-based paradigms
The social behavior-based paradigm models the cooperative relationships of intraspecific individuals.In 1989, Beni and Wang [99] first proposed swarm intelligence in the context of cellular robotic systems and considered swarm intelligence as the process of combining computation and dynamics.Subsequently, swarm intelligence was extended to the collective and social behavior of insects and other animals, bionic "nature" (i.e., physical or biological systems) to build adaptive, decentralized, flexible, and robust artificial systems as a paradigm for problem solving [100].
According to the forms of collective animal or plant behavior, social behavior-based paradigms can be classified as foraging, navigation, parasitism, reproduction, competition, and others.The main characteristics of paradigms include self-organization, indirect communication, high parallel distribution, stability, and adaptability [101].This family of paradigms is composed of a population of agents capable of interacting with each other or with the environment, where the agents interact cooperatively by using simple local rules in a decision space to accomplish a certain task [14].Figure 6 presents a general framework based on the social behavior-based paradigm, with the update step of the PSO algorithm as an example.The diversity of social behavior-inspired paradigms reflects the complexity and richness of the natural world.These paradigms, inspired by the collective behaviors of social animals or plants, offer unique perspectives and strategies for problem solving.However, as highlighted by the "no free lunch" (NFL) theorem, no single paradigm excels in all situations.We summarize the strengths and weaknesses of these paradigms in Table 1.Obviously, these methods inspired by different behaviors induce similar adaptive mechanisms to balance exploration and exploitation in optimization to improve algorithm performance.However, most paradigms still suffer from common shortcomings such as parameter sensitivity and limited theoretical explanation.This section aims to summarize the natural mechanisms and update rules of social behavior-based paradigms, thus providing a substantial contribution to the reduction of unreasonable "metaphors".

Foraging behavior
Group foraging, also known as social foraging, is the process of finding, capturing, and feeding by a group of closely allied individuals, usually in the form of collective movements.The success of foraging relies on the individual itself and the group.In group foraging, members of groups such as flocks of birds, wolves, and elephants may benefit individual foragers by increasing foraging time, food contact rates, and capture efficiency and reducing the unit of time or energy expended of one individual [102,103].
The main factor that distinguishes the collective behavior of group foragers from that of individual foragers is group cohesiveness.Generally, in social foraging theory, the problem involves the fitness of an individual's behavior, and the model is based on game theory and evolutionary stabilization strategies [104].As noted by studying male rhesus macaques in [105], the marginal value theory (MVT) is the most appropriate for observing group foraging behavior.The MVT theory suggests that food resources in an environment are distributed in discrete patches, where the animal consumes food resources by feeding in patches and moving between patches.The animal is faced with diminishing returns as the food in a patch increases in time, so it has to choose the right time to leave to feed in the next patch.According to this theory, group foraging is a random rather than a deterministic process, and information is also "shared" between groups.In this process, individuals may be forced to search and forage, which is a dynamic exploitation of resources [106].
By studying the habits of a number of group foragers, and combining the group foraging characteristics described above, a number of swarm intelligence models based on group foraging behavior have been proposed, such as PSO [107], ant colony optimization (ACO) [108], artificial bee colony algorithm (ABC) [109], and grey wolf optimizer algorithm (GWO) [110].These models typically cover the 3 phases of group foraging: locating and surrounding prey, hunting, and attacking, aiming at "information sharing" in a stochastic environment with dynamic "exploitation-exploration" to find the optimal solution [103].
Among these paradigms, the top 4 cited paradigms are PSO, ACO, ABC, and GWO.The relevant features are shown in Table 2.The parameter introductions related to Table 2 can be found in Tables 3 and 4.
The PSO algorithm, the best-known biological-based paradigm, was originally proposed for modeling the social behavior of graceful but unpredictable movements of a flock of birds, where each agent serves as a collision-proof bird in flight.Essentially, simple social interactions are utilized to generate flock intelligence in the PSO algorithm [111].According to Table 2 and Fig. 6A, the PSO update principle consists of 2 main components, velocity update and position update.The velocity of each particle v (t)  i,d is adjusted by considering its own bestknown position (P best, i ) and the best-known position among all particles in the group (G best ).This update contains cognitive and social components, where cognitive refers to the particle's own experience and social refers to learning from the group.After updating the velocity, the position of each particle x t i,d is updated by adding its new velocity to its current position.This step moves the particle to a region of higher fitness in the search space.Overall, the PSO algorithm guides them toward optimal solutions based on their own experiences and the experiences of their neighbors in the swarm.The ACO algorithm is inspired by the foraging behavior of ants, which use pheromones to communicate and share information.Table 2 illustrates that the main function of ACO involves 2 main steps: (a) Edge selection: Each ant selects the next node to visit based on a probability that depends on the pheromone level and heuristic information.(b) Pheromone update: After all ants have completed their tours, pheromone levels on edges are updated to reflect the quality of the solutions found, with pheromone evaporation and deposition guiding future search efforts.The ABC algorithm models the nectar collection mechanism of bees by dividing the colony into employed bees, onlookers, and scouts following a social hierarchy.Table 2 shows that ABC uses employed and scout bee phases for updating solutions.Employed bees search for new solutions by exploring the neighborhood of their current positions.Scout bees introduce diversity by randomly generating new solutions when an employed bee's solution cannot be improved further.This dual approach balances exploration and exploitation, effectively preventing stagnation.
Similar to the ABC, grey wolves also have a social hierarchy.According to Table 1, the update mechanism of GWO emulates grey wolves' social hierarchy and hunting tactics.Wolves (agents) encircle prey (best solution) and update their positions accordingly.GWO categorizes wolves into alpha (x α , the best solution), beta (x β , the second-best), and delta (x δ , the thirdbest).The remaining wolves in the pack update their positions based on these 3 leaders.

Navigation behavior
Several species of animal in nature move in an orientated manner seeking food, avoiding predators, finding mates, homing, and navigating between areas essential to their survival.The role of navigation in animals is of great significance, as it involves the neural processing of sensory inputs, and the integration of different types of cues to orientate oneself and locate direction, aiming to guide toward their destination [112].In 1873, Darwin [113] proposed that animals have the ability to navigate by dead reckoning, i.e., navigating by magnetic "compass" perception or by stars.From then on, researchers have investigated the mechanisms of how animals navigate based on senses.Existing relevant mechanisms and hypotheses have been proposed such as sun-based orientation, magnetism-based orientation, olfactory-based navigation, and cognitive map-based navigation.Generally, animals possess more than one orientation mechanism [112].The researchers established prevalent optimization algorithms based on the navigation behavior, as shown in Table 5.The parameter introductions related to Table 5 can be found in Tables 3 and 6.
For both diurnal and nocturnal animals, celestial cues provide a wealth of information about orientation.A typical example is the moth.The moth has evolved to travel through night adopting a mechanism of transverse orientation to navigate.Since moonlight is at a great distance, this mechanism allows moths to fly in a spiral by maintaining a steady angle with respect to the moon.As shown in Fig. 7, in the case of artificial light, moths attempt to move in a spiral at the same angle as the light source.Inspired by the navigation mechanism of moths, Mirjalili [114] proposed the moth flame optimization (MFO) algorithm.According to Table 5, MFO first generates a random population of moths in the solution space and then calculates the fitness value of each moth to determine their positions.The best position found so far is marked by the flame.The algorithm relies on a spiral motion function to guide the moths toward the flame, updating their positions accordingly.This process is repeated iteratively, updating the positions of the moths and flames until the termination condition is satisfied, ensuring that the moths converge toward the optimal solution.
Bats are the only flying mammals.They make extensive use of echolocation.They emit loud pulses of sound and listen for echoes bouncing off their surroundings.Yang and He [115] presented a bat algorithm (BA) by modeling the dynamic behavior of bats with a type of ultrasound to detect prey and avoid obstacles.According to Table 5, bats in the algorithm have distinct positions, velocities, frequencies, loudness, and pulse rates.Initially, bats randomly navigate the search space, with frequency updates guiding velocity and movement per iteration, akin to echolocation.Frequency adjustments based on proximity to targets enable switching between exploration and exploitation.As bats near-optimal solutions, they reduce the loudness and increase pulse rates, similar to natural bat behavior when targeting prey.This behavior ensures that the bats focus on fine-tuning their positions as they approach the optimal solution.
Navigational behavior is the foundation of biological migratory activity.Studies have shown that seagulls establish an olfactory map with scent to navigate and migrate.Most of the seagulls migrate seasonally, heading in groups for rich food sources and habitats [116].In addition, seagulls also attack migrating birds [117].Dhiman and Kumar [118] proposed the seagull optimization (SOA) algorithm inspired by the migration behavior and attack behavior of seagulls.Migratory  2) Few parameters.
3) Theoretical foundation in routing and scheduling.
2) High computational demand for complex or large-scale problems.
1) Slower convergence in complex scenarios.
2) Robust parameter changes due to the simple parameter and spiral movement.
3) Ineffectiveness in discrete problems.4) Difficulty with dynamic problems.BA [115] 1) A good combination of major advantages features of PSO and harmony search.
2) High convergence rate in the early stages of the iterations.
3) Good exploitation ability 1) Risk of premature convergence the exploration and exploitation.
2) Diverse search mechanisms include spiraling flight for local search and straight flight for global search.
1) The exploration ability is relatively weak.
2) Convergence speed can be slower in simpler problems.
2) Effective in dynamic environments.
1) Limited testing and validation.
2) Collaborative search balance exploration and exploitation.
3) Scalability challenges in complex scenarios. (Continued) behavior affects global exploration ability, and aggressive behavior influences local exploitation ability.As shown in Table 5, the optimization steps of the SOA are primarily divided into 2 phases: migration and attack.During the migration phase, agents initially avoid collisions and then move.During the migration phase, agents initially avoid collisions and then move according to the global best position found so far.In the attack phase, seagulls exhibit spiral natural movements, thereby updating their positions.Apart from seagulls, monarch butterflies are known for migrating on a large scale by using compass navigation.Wang et al. [119] established a monarch butterfly optimization (MBO) algorithm by modeling the migration behavior of monarch butterflies.In MBO, the position of each monarch butterfly represents a feasible solution.Each monarch butterfly will be distributed over 2 continents ("Land" and "Subland") to search for the optimal solution through offspring migration and computing fitness.According to Table 5, the main update involves migration and adjusting the position of the butterflies.For the butterflies in the "Land", the positions are updated using the migration operator.For the butterflies in the "Subland", use Lévy flight behavior or other modes to update their location.The migration operator allows butterflies on the "Land" to update their positions based on a probabilistic mechanism, enhancing global exploration.The adjusting operator, incorporating Lévy flight behavior, enables butterflies on the "Subland" to adjust their positions, facilitating local exploitation and fine-tuning of solutions.
Homing refers to the ability of an animal to navigate back to its original location in an unfamiliar area, which is found in birds, salmon, burrows, and other animals.The pigeon is a well-known animal with homing behavior accomplished through navigation and is widely used in wartime communication, daily posts, etc.The pigeon forms a "navigation map" through mechanisms such as memorized visual landmarks, compass, and magnetic field [120].During homing, the pigeon constantly evaluates its route to navigate in the correct direction through multiple mechanisms.Inspired by the homing of pigeons, Duan and Qiao [121] established a pigeon-inspired optimization

Paradigm
Characteristics and strengths Weakness CSO [142] 1) Dynamically switch between seeking and tracing modes.
2) Dynamically balance exploration and exploitation.
3) Robust to noise and user-friendly simplicity.
1) Risk of local minima traps.
Influenza predicting [288] Parametric optimization [289] Scheduling problems [290] Energy system [291] Ant colony optimization (ACO) [108] Group foraging and trail pheromone Edge selection: Scheduling problems [292] Traffic congestion problems [293] Network routing [294] Artificial bee colony algorithm (ABC) [109] Group foraging waggle dance and social hierarchy Employed bee phase: Assignment problems [295] Image and video processing [296] Portfolio selection [297] Grey wolf optimizer (GWO) [110] Group foraging and social hierarchy Encircling prey: x (t+1 Feature selection [298] Power dispatch problems [299] Scheduling problems [300] (PIO) algorithm based on the map and compass mechanism of the pigeon homing behavior.PIO achieves the best position by constructing and updating the "map and compass operator" and "landmark operator" using the pigeons' locations throughout the iterations.According to Table 5, when iteration t ≤ t 1 max , pigeons use the map and compass operator to update their positions, where t 1 max is a custom parameter.When iteration t 1 max < t ≤ t max , pigeons use the landmark operator to update their positions.This dual-phase update mechanism ensures that the pigeons effectively balance exploration and exploitation.

Parasitic behavior
Parasitism is the phenomenon when 2 organisms live together, one benefiting while the other suffers.The former is called the "parasite" and the latter the "host", with the latter providing nutrients and shelter to the former.About 40% of known species are parasitic, and almost all animals are hosts to at least one parasite [122].There are more than 400 types of parasites in humans [123].The Red Queen hypothesis proposes that both hosts and parasites are constantly interacting, opposing, and fighting each other in a parasitic and antiparasitic struggle, with both co-evolving to produce diversity [124].The way birds nest parasitically is a compelling example.
Bird brood parasite is a special breeding behavior whereby organisms lay their eggs in the nests of other birds, leaving the others to incubate and raise them.The most famous example of nest parasitism in birds is the cuckoo, which possesses a superb method in the art of deception.It removes one of the eggs laid by its host and then lays it itself.The main factor in the selection of phenotype for cuckoo parasitic eggs is the ability to successfully bypass host defenses, that is, careful matching by mimicking the pattern and color of the host egg, a skill that requires a high degree of accuracy to ensure success [125].This process pays off after a while; the cuckoo's eggs will hatch before the host's, confusing the host into instinctively driving its own eggs out of the nest and thus increasing care for the cuckoo's chicks.Cuckoo chicks are naturally cunning and will mimic the calls of their host chicks to gain more foraging opportunities.On the other hand, if a host recognizes a cuckoo's eggs in its nest, either it throws them away or keeps its own eggs to build a new nest.Hence, the cuckoo must imitate the host's eggs more accurately, while the host must improve its skills in identifying the parasitic eggs, which is called the struggle for survival [126].
Yang and Deb [127] developed a cuckoo search (CS) algorithm mimicking the egg-laying pattern of the cuckoo, incorporating the typical features of Lévy flight with a power-law pattern exhibited by the flight behavior of many animals and insects.In this algorithm, each cuckoo lays one egg (solution) at a time and dumps it in a random nest, and the best eggs will be taken to the next generation.The main idea of the update step in the algorithm is: where for each cuckoo i, x i represents the nest position.α is the stepsize control quantity.Lévy(λ) is derived from Lévy flight function.Moreover, CS is widely applied in frequency control problems [128], traveling salesman problems [129], and energy power [130].

Competitive behavior
Darwin [38] emphasized the importance of competition as a universal principle of biology.Biological competition is a dynamic process, divided into intraspecific and interspecific competition.Intraspecific competition occurs when members of the same species compete for the same resource, and interspecific competition is probable to happen when individuals of 2 different species share a limited resource.Models such as the "competition Lotka-Volterra equation" mathematize the effects of dynamic interspecific competition on populations [131].Generally speaking, competitive behavior among animals is often linked to aspects such as access to mates, food, territory, or valuable (1)

P best, i
The personal best position of agent i has found so far.

G best
The global best position found by the entire swarm.resources.Thus, animal competitive behavior is intricately woven into various aspects discussed in other parts of the "Biological-Based Paradigms" section.Interestingly, competition is evident in plant species, particularly in weeds.Weeds are vigorous, aggressive growth habit plants that adapt rapidly to changes in their environment resulting in a threat to agriculture.The biodiversity of weeds provides plants with diverse characteristics to adapt and exploit opportunity spaces, and to adapt locally through natural selection.As population density increases, it becomes more difficult for species with lower fitness to survive.
Inspired by the competitive relationship between invasive weeds and the native plants, Mehrabian and Lucas [132] proposed an invasive weed optimization (IWO) algorithm to solve the solution for optimization through weed initialization, growth, reproduction, spatial distribution, and competition.The main idea of the update step in IWO is reproduction and spatial distribution.For each seed i, S i new seeds are generated, where S i is proportional to the seed's fitness: where S max and S min are the maximum and minimum number of seeds, respectively, and gen is the current generation number.The seeds are dispersed using a normally distributed random number with a mean of zero and a standard deviation that decreases over generations: where σ initial and σ final are the initial and final standard deviations, respectively, and n is a nonlinear modulation index.In summary, IWO is a powerful and flexible algorithm inspired by the colonizing behavior of weeds.IWO is typically employed in engineering design [133] and recommended system [134].

Reproductive behavior
Sexual selection is a mode of natural selection by which animals in nature often obtain the right to mate.Darwin [135] stated that sexual selection depends on the struggle between males for the right to approach females.There are 2 mechanisms of sexual selection: intra-sexual selection, which is a competition between members of the same sex (usually males) to obtain a mate, and inter-sexual selection, in which members of one sex (usually females) select members of the opposite sex [135].For most species, mating takes place between 2 individuals of the opposite sex, and animal courtship behavior is also a manifestation of sexual selection.The presentation of animal courtship behaviors involves complex displays of dancing, vocalization, nesting, or fighting ability, designed to attract the attention of the opposite sex.Males in many species produce complex multimodal signals covering more than one sensory modality, such as a combination of tactile, visual, and auditory signals.ρ Rate of pheromone evaporation (a constant between 0 and 1).This parameter controls the rate at which pheromone evaporates from the path.A higher value of ρ means faster evaporation.A random number between [−1,1], controlling the perturbation and influencing the direction and magnitude of the change.
The current solution for a randomly chosen kth agent in the dth dimension at iteration t.The kth agent is selected randomly and must be different from the ith agent.
x p The position of the swarm (or an estimated position based on the best solutions found so far) at iteration t.A A coefficient vector that influences the step size and direction of the wolves' movement.

D
A coefficient vector that influences the oscillation behavior around the swarm.
x α , x β , x δ Best 3 solutions obtained so far.
behaviors, with some typical examples presented in Table 7.The parameter introductions related to Table 7 can be found in Tables 3 and 8. Male courtship behavior is common, for example, male satin blue gardeners construct gazebos to attract females.They compete by stealing other males' ornaments and will destroy their gazebos [136].Building on these observations, Moosavi and Bardsiri [137] proposed a satin bowerbird optimization (SBO) algorithm to model the courtship behavior of male satin blue garden birds.SBO uses the "gazebo" as a solution, which updates the gazebos with information sharing and destroys other gazebos for mutation.According to Table 7, SBO incorporates elitism and mutation to update solutions.Elitism preserves the best solutions akin to the most attractive bowers built by experienced male bowerbirds.Mutation mimics the unpredictable environmental factors affecting bower construction, ensuring a degree of robustness and adaptability in the algorithm by occasionally altering bower (solution) attributes.
Mayflies are another example, with most males gathering in swarms a few meters above the water to perform dances by moving in up-and-down patterns.Females fly into these swarms and mate with the aerial males.Mating may last only a few seconds, and when complete, the female drops her eggs to the surface.Inspired by this, Zervoudakis and Tsafarakis [138] proposed the mayfly algorithm (MA), where the position of each mayfly in the search space represents the potential solution, and the mayfly continuously adjusts its own position during flight to obtain the optimal solution.According to Table 7, the MA uses 2 key phases: nuptial dance (exploration) and mating (exploitation and crossover).The nuptial dance phase updates the position of the male mayflies and female mayflies with a different formula.In the mating phase, crosses between male and female mayflies produce new offspring (solution).

Migration
Migration operator: Adjusting operator: Wireless sensor network optimization [312] Schedule problems [313] Energy management and conversion [314] Pigeon-inspired optimization (PIO) [121] Homing, map, and compass mechanism Map and compass factor phase: Landmark factor phase: Image process [315] Unmanned aerial vehicle (UAV) flocking control [316] Many-objective optimization problems [317] Control parameter design [318] points of the web with pheromones to attract males.They eat males in mating.Producing offspring also eat each other and may even eat their mothers.Inspired by the reproduction behavior of black widow, Hayyolalam and Kazem [139] developed the black widow optimization (BWO).Potential solutions are considered as black widow spiders, allowing healthy and strong individuals to survive.According to Table 7, BWO generally consists of 3 main processes: mating, cannibalism, and mutation.The mating process involves the crossover operation, cannibalism is related to the selection mechanism, and mutation introduces randomness and diversity.

Other behaviors
Beyond the social behaviors described above, there are still some popular paradigms that also model other characteristics and behaviors of species as shown in Table 9.The parameter introductions related to Table 9 can be found in Tables 3  and 10.
The coyote optimization algorithm (COA) [140] models the social organization of North American coyotes and their living conditions.According to Table 9, the position of each coyote in the search space is updated based on its social condition and the social condition of other coyotes in its pack and neighborhood.The elephant herding algorithm (EHO) [141] mimics the herding behavior of elephants, specifically how they live in clans and use their wisdom to guide the herd.According to Table 9, each clan's position (solutions) updates via the matriarch's guidance (x ci, best ).A portion of the least effective elephants become nomads, exploring solutions independently.The cat swarm optimization (CSO) algorithm [142] is based on the search and tracing behaviors of cats.CSO randomly classifies the cats into seeking and tracing modes according to a mixture

cos(2πl)
A cosine function that introduces oscillation around the flame.

A t
Average loudness of all bats at iteration t. f i Frequency associated with the ith bat, influencing step size.f min and f max Minimum and maximum frequencies.
It is responsible for balancing the exploration and production capacity of the algorithm.

M t i
The migration vector for the ith seagull at iteration t.
s r represents the radius of the spiral.k is a random number in [0, 2π].u and v represent the constants of the helix shape.

N Total number of a population. peri
A period factor that determines the likelihood of migration.
The position of 2 monarch butterflies randomly selected from subpopula-tion1 (r1) and subpopulation2 (r2) in the dth dimension at iteration t.
The position of 2 monarch butterflies randomly selected from subpopula-tion2 (r2) in the dth dimension at iteration t.BAR A threshold value used in the adjusting operator to determine different adjustment behaviors.
α A scaling factor used in the Le ′ vy flight term.p A probability threshold for determining migration or adjustment actions.

R
A constant that controls the rate of decay for the velocity.
x t center The weighted average center position of all positions at iteration t.

Immune system
According to modern medicine, the immune function is a response to antigenic stimuli, which is expressed in the ability of the IS to identify itself and to eliminate non-self [144].The IS serves as the defense system of organisms.Compositionally, the IS consists of immune organs, immune cells, and immune molecules.The immune organs such as the spleen and the thymus are responsible for the production of immune cells.Immune cells are cells involved in the immune response process, such as lymphocytes and phagocytes, while immune molecules are mostly substances secreted by immune cells, such as antibodies and complement [145,146].
The IS has a hierarchical defense mechanism, commonly divided into 3 layers: (a) physical, (b) innate immunity, and (c) adaptive immunity [145].The first layer consists of physical and chemical barriers such as epidermal skin and mucous membranes.The second layer is the immune response generated by the innate IS, which allows a rapid response to a wide range of pathogens, such as phagocytosis and the complement system.Layers (a) and (b) are innate immune mechanisms present in all multicellular organisms and rely on germline-encoded receptors from the innate IS to recognize pathogens.Layer (c) employs variable antigen-specific receptors generated by gene fragment rearrangements.Adaptive immunity has been shown to be a major driver of selection for tumor suppressor gene inactivation [147].Adaptive immunity occurs when the IS recognizes invading pathogens through a variety of responses.Following pathogen clearance, some of the immune cells may become memory cells and remain in the body for a long time.
The adaptive IS includes humoral and cellular immunity, with B lymphocytes primarily involved in the humoral immune process and T lymphocytes noted to be involved in the cellular immune process.As shown in Fig. 8, when being activated, some B lymphocytes and T lymphocytes become memory B cells and memory T cells, respectively.When the same pathogens reinvade, the adaptive IS rapidly produces a strong adaptive immune response [148].Innate and adaptive immunity interact functionally in modern vertebrates.However, innate responses occur to the same extent regardless of the exposure to infectious agents encountered, whereas adaptive responses improve with repeated exposure to specific infections.Adaptive immunity can be acquired through natural infection or artificial Mutation: Congestion management [319] Engineer design [320] Mayfly algorithm (MA) [138] Reproductive waggle dance Nuptial dance phase: For male: x t+1 male,i = x t male,i + v t+1 i For female: Training ANNs [321] Energy modeling [322] Black widow optimization (BWO) [139] Reproductive siblicide and pheromones Mating (crossover): Image processing [323] Autonomous car-driver scheme [324] vaccination.The genealogical relationship between memory and effector cells has profound implications for vaccine design and the development of effective T cell-based therapies.

Artificial immune system
Biological ISs possess powerful information processing capabilities with feature extraction, learning memory, fault tolerance, and distributed properties [149,150].Therefore, motivated by the biological IS, researchers have developed the AIS.The basic AIS models antigen recognition, memory, and selfregulation, which is akin to the biological IS.As shown in Fig. 9, AIS considers the antigen as the problem to be solved and the antibodies as the potential solutions.Unlike fitness evaluation in EAs, AIS employs affinity as evaluation, including antibody-antigen affinity as well as antibody-antibody affinity.The affinity evaluation metric reflects the diversity of ISs.AIS is highly adaptable and robust.It is not sensitive to the setting of algorithm parameters or the quality of the initial solution.Moreover, centralized control is not required and can be processed in parallel, making AIS particularly suitable for multimodal optimization problems.
As shown in Table 11, AIS has been developed around 5 main theories of immunology: clonal selection theory, immune networks theory, negative selection theory, immune danger theory, and vaccination theory.
Clonal selection theory.Clonal selection theory explains the mechanism by which lymphocytes respond to specific antigens [151,152].Figure 10 illustrates a simple diagram of clonal selection theory.Antibody-forming cell precursors, especially B cells, undergo T cell-dependent activation or T cell-independent activation and are clonally selected to produce antibodies.During this process, B cells undergo affinity maturation, a Darwinian evolution process characterized by B cell mutation and selection, which ensures that only those B cells producing highaffinity antibodies survive.These specialized cells then clone and differentiate into plasma cells and memory B cells.Plasma cells continuously secrete antibodies, making the IS more efficient at recognizing and clearing pathogens.Memory B cells have the unique ability to recognize previously encountered antigens, allowing for a more rapid and effective immune response upon reinfection [153].
The clonal selection algorithm is a class of algorithms obtained by mimicking the doctrine of clonal selection theory.A representative example is CLONALG [154].CLONALG mimics the affinity maturation process in the immune response, which includes basic strategies such as selection, proliferation, and mutation.Determines the attraction power in the goal bower.
x j, d The position of a selected bowerbird by the roulette wheel procedure in the dth dimension.x elite, d The position of the elite (best) bowerbird in the dth dimension.
The standard deviation controlling the spread of the mutation.
λ A random number or a crossover factor that controls the contribution of each parent to the offspring.r A random number following a specific distribution, which could be uniform or Gaussian.

r g Cartesian distance between x i and G best . r p
Cartesian distance between x i and P best .r mf Cartesian distance between male and female mayflies.
x t male,i,d Current position of the ith male mayfly in the dth dimension at iteration t.
x t female,i,d Current position of the ith female mayfly in the dth dimension at iteration t.
x offspring Position of the offspring.
f l A scaling factor for the random component.
The position of the ith male parent at iteration t.
The position of the ith female parent at iteration t.
Artificial immune network theory.Immune network theory attempts to explain how the adaptive IS regulates itself.The related work was mainly proposed by Niels Jerne in 1974.The immune network theory suggests that IS can maintain immune memory through a mutually reinforcing network of B cells.Additionally, antibodies, like other molecules, have antigenic epitopes that can be recognized by other antibodies, allowing for the distinction and regulation of different antibody types.The core idea of the theory holds that the elements of the IS (cells, antigens, antibodies, etc.) do not exist in isolation.They have a relationship of mutual stimulation, restraint, and recognition [150,155].
The artificial immune network algorithm is inspired by immune network theory.There are 2 main mainstream artificial immune network models, the resource-limited AIS (RLAIS) [156] and the AiNet [157].RLAIS introduces the concept of an artificial recognition ball (ARB), which functions similarly to B cells.The AIS consists of a fixed number of ARBs that are stimulated by the primary stimulus ps of the antigen, the stimulus nn of the adjacent antibody, and the inhibitory ns of the adjacent antibody.The degree of antibody cloning is determined by the stimulus to the ARB.The AiNet models the immune network's response to antigenic stimuli, incorporating processes such as antibody-antigen recognition, immune clone proliferation, affinity maturation, and network inhibition.
Negative selection theory.T cells recognize self and non-self antigens through T cell receptors (TCRs), which have a variety
x p, avg Average position of all coyotes in the pack.
x ci, k Position of kth elephant in cith clan.

x ci, best
Best elephant in the cith clan.

x center, ci
Center position in the cith clan.

N ci
Number of elephants in the cith clan.SRD Seeking range of dimension, determines the local search range.
Attractiveness function, which typically depends on the distance r between fireflies i and j. β 0 is the initial attractiveness at r = 0 and γ is the light absorption coefficient. α Step size coefficient.
The position of the jth firefly at iteration t, which is more attractive (has better fitness) than the ith firefly.Resource limited artificial immune system [156] AiNet model [157] Classification problem [339], Disease detection [340] Negative selection theory Negative selection algorithm [159] Intrusion detection systems [341], Fault detection system [342]

Immune danger theory
The dendritic cell algorithm [161] Anomaly detection [343], Error detection [344] Vaccination theory Immune genetic algorithm [163] Path planning [345], Anomaly detection [346] of structures resulting from gene rearrangements.Antigenpresenting cells (APCs) capture antigens and break them down into small peptides that interact with the TCRs.The affinity between antigens and TCRs, determined by their structures, controls T cell activation [158].To prevent autoimmunity, T cells that recognize their own antigens engage in clonal deletion in the thymus [158].Thus, T cells that recognize the "self " are eliminated, while those that do not recognize the "self " mature and are used to recognize the "non-self ".Forrest et al. [159] developed a negative selection algorithm for anomaly detection based on positive and negative selection, which is similar to the process of "negative selection" that T cells undergo during maturation.In this algorithm, detectors are randomly generated, those detecting themselves are deleted, and those detecting non-self are retained for anomaly detection.
Immune danger theory.The core idea of the danger theory [160] is that the IS distinguishes between danger and safety by recognizing pathogens or signals from injured or stressed cells and tissues.Danger signals, which are crucial determinants of the immune response, activate APCs upon detecting damage.In contrast, healthy cells or cells undergoing normal physiological death do not emit danger signals.Any intracellular substance released from damaged or injured cells can act as a danger signal.
Dendritic cells, which are immune cells involved in antigen presentation, play a key role in this process.By integrating the immune danger theory into AIS, Aickelin and Greensmith [161]  Vaccination theory.Vaccines are biological therapies that provide adaptive immunity to specific infectious diseases.Modern vaccines usually contain components that resemble the diseasecausing microorganism, such as weakened or killed microorganisms, their toxins, or a protein on their surface.The vaccine is administered to produce memory cells that generate antibodies corresponding to the pathogen.Additionally, T cells can destroy the virus's ability to replicate by seeking out and destroying infected cells [162].
Inspired by this process, Jiao and Wang [163] proposed a new GA based on "immune vaccination", namely, the immune genetic algorithm (IGA) that converges with probability 1. IGA constructs immunity operators through 2 main steps: a vaccination and an immune selection.The convergence speed of IGA is improved by introducing immunity operators to prevent the degradation of population diversity.

Social-Cultural-Based Paradigms
Social-cultural-based paradigms are inspired by human social and cultural patterns of behavior.During the development of modern society, many behaviors containing human prior knowledge have been inherited.These prevalent social-culturalinspired paradigms are detailed as shown in Table 12, including brain storm optimization (BSO), cultural algorithm (CA), imperialist competitive algorithm (ICA), and teaching-learningbased optimization (TLBO).The parameters related to Table 12 are shown in Tables 3 and 13.

Brain storm optimization
Brain storm [164] occurs in human interaction and cooperation.When faced with a problem, a bunch of people with different knowledge backgrounds cooperate and communicate with each other.Then, the problem can be solved with high probability.Unexpected wisdom is born in this process.During the brainstorming process, there usually exists a facilitator, a brainstorming group, and several owners of problems that need to be solved.Facilitators force the brainstorming group to generate ideas based on certain principles.The problem owner selects better (and noteworthy) ideas from the set of ideas generated.Inspired by the brainstorming process, BSO [165] mainly consists of clustering and mutation.Through the convergence and divergence operations, the individuals in the population are grouped and diverged in search space.The optimal solution is searched during the aggregation and dispersion process.Because brainstorming organically combines swarm intelligence and data mining, it has been widely used in power systems, aerospace, economics [166], and other fields.

R
A random number following a specific distribution, which could be uniform or Gaussian.T Maximum number of iterations.
x l , x u Lower and upper bounds of the independent variable in excellent individuals, respectively.

Cultural algorithm
Culture [167] is a system of conceptual phenomena socially and historically encoded within and between groups of symbols.In recent years, some works have modeled the process of cultural evolution from both the perspectives of micro-evolution (the transmission of behavior or traits among individuals in a group) and macro-evolution (the formation of generalized beliefs based on personal experience).These generalized beliefs are used to constrain the behavior of individuals in related groups.The cultural system of dual inheritance described above supports the transmission of information at the individual and group levels.
Inspired by the evolution process of culture, a 2-layer evolutionary mechanism consisting of population space and belief space is designed in the CA [168].Population space models the evolution process of biological individuals according to certain behavioral rules from a microscopic perspective, and belief space models the evolutionary process of cultural formation, transmission, and comparison from a macroscopic perspective.The 2 spaces relate to each other based on the communication protocol, which effectively extracts and manages the evolutionary information.

Imperialist competitive algorithm
Beginning in 1970, the developed countries tried to dominate the less developed countries politically and militarily to expand their power and plunder their resources, which is called modern colonialism in history.The competition between imperialists entails the prosperity and advancement of the dominant national economy.To facilitate the dissemination of their values, the infrastructure of the colony was further built [169].From an optimization perspective, the phenomenon of imperialist competition can be explained as follows: Colonies are lifted out of the valley (current position) and pushed to the peak of imperialism (new minima).The new status of the colonies could be better than imperialism at any time.The movement of the economic axis meant that the colonies improved their economic conditions by being influenced by the imperialist economy [170].In ICA, those of the best countries (lower cost) are chosen as imperialist countries and the rest are treated as colonies, as shown in Fig. 11.After colonies were carved up by the imperialist states, they moved toward their associated imperialism in the space of the cultural state.That is to say, whether an empire survives or not hinges on its ability to undertake colonies from other competitors.Great empires grew in strength, while weaker empires crumbled.ICA has been broadly used in scheduling problems [171].

Teaching-learning-based optimization
The TLBO [172] is inspired by the wisdom of the classroom teaching process, which mimics the teaching process of a teacher to a learner in a class.The improvement of the level of the students requires the teacher to "teach".Students need to "learn" from each other to promote the absorption of knowledge at the same time.During the teaching process [173], learners are viewed as points distributed in the decision space, corresponding to a population of evolution-inspired paradigms.The bestperforming students are defined as teachers of the class.A teaching phase and a learning phase are designed in the TLBO.
The learner not only improves his own knowledge level by learning from the teacher in the teaching stage to improve the average knowledge level of the class but also randomly learns from other learners in the learning stage to broaden his knowledge level.
The TLBO has been applied to many industrial fields such as scheduling and transformer fault judgment [174].

Others
In addition to the above approaches, there is a range of paradigms inspired by culture and society.An important point of Confucius is that moderation is the best rule.Criss-crossing [175] is a new search paradigm inspired by the Confucian doctrine of the mean and the crossover operation in GAs.Many aspects of the universe are governed by duality, which refers to 2 opposing forces or conflicting states of nature at work.In Chinese philosophy, this idea is described as "yin" and "yang", 2 complementary and interdependent extremes.One aspect gradually changes the other.This process is repeated until the balance of these 2 aspects produces harmony.Yin-yang pair optimization [176] is inspired by this idea to expect to balance the relationship between exploration and exploitation in optimization.The ultimate goal of a business hierarchy is to accomplish business-related tasks in the best possible way.Heap optimization [177] employs the heap structure to model the hierarchical structure of the company.The concept of the heap is adopted to form interactions between individuals.Three mathematical models are constructed for new individuals.Humans are great imitators or followers when solving any task.Groupsolving skills are more effective than individual-solving skills when developing and exploring given problems.Social group optimization [178] is inspired by this idea so that each person enhances their knowledge by communicating with others in the group and learning from the best in the group.The above methods all hope to provide more advanced tools to solve practical problems by leveraging the phenomena in society and culture.

Science-Based Paradigms
Science-based paradigms are inspired by proven scientific theorems.This formulated knowledge comes from a variety of disciplines including natural, social, and formal sciences.These science-inspired paradigms are described in detail in this section, mainly in 3 areas: physics, geography, and mathematics as shown in Table 14 and Fig. 12.

Physics
Simulated annealing (SA) [179] is derived from the principle of solid annealing.The solid is heated to a sufficient height and then slowly cooled.During heating, the internal particles of the solid become disordered as the temperature rises.The internal energy increases.Then, particles tend to be ordered gradually, reaching an equilibrium state at each temperature.Finally, the particles achieve their ground state at room temperature.Moreover, their internal energy is minimized.The slow cooling implemented in the SA algorithm is explained as that the probability of accepting a worse solution slowly decreases as the solution space continues to be explored.SA algorithm is a general optimization algorithm.

Mathematics
Sine cosine algorithm Gradient-based optimizer machine learning [182].Recently, SA has been used to optimize the tin oxide/MoS 2 -based Boltzmann machine [183] as shown in Fig. 13.By adjusting the value of T eff , different "cooling" strategies can be obtained.The figure shows the performance of 4 different strategies for optimizing the Boltzmann machine, including high T eff to low T eff , low T eff to high T eff , low T eff , and high T eff .
Gravity refers to the tendency of objects to accelerate each other.Each particle of the universe gravitates to others.Since gravity is everywhere, it makes it unique from other natural forces.Newton's law of universal gravitation implies that gravity works among the separated particles with no intermediary or delay.Every particle attracts other particles with gravity.Inspired by this, the search agent for the gravitational search algorithm (GSA) [184] is a set of masses based on the interaction of Newton's laws of gravity and motion, which mainly guides the motion of each particle according to Newton's law of universal gravitation between 2 objects.The gravitational force is proportional to the mass of 2 particles and inversely proportional to the distance between them.GSAs have been widely used in power engineering [185] and control systems [183].
Inspired by quantum computing, quantum evolution (QE) [186] employs qubits to encode chromosomes, which mainly includes quantum chromosome observations and quantum gate updates.Quantum revolving gates, which play a critical role in QE, are the most commonly used quantum gates in this process.Because of the significance of revolving gates, they have received extensive attention in QE [187].The simplicity and scalability of the QE structure have made it popular to integrate with various other heuristics, including PSO [188], immune cloning algorithms (ICA) [189], SA [190], and so on.Recently, Chai et al. [191] proposed the shortcuts to quantum approximate optimization algorithm (S-QAOA), which is an ideal choice for solving combinatorial optimization problems using current noisy quantum computers.
The Big Bang theory states that the universe began with a massive expansion known as the big bang [192].Related to this hypothesis is that the big bang is the foundation of everything in nature.The multi-verse optimizer (MVO) [193] is modeled based on the principle that matter in the universe is transferred from white holes to black holes through wormholes.In the random creation process of the universe, objects with high expansion rates always tend to objects with low expansion rates due to gravitational effects.This gravitational effect can make objects transfer.According to the relevant cosmological rules, objects can gradually tend to the optimal position in the search space.MVO has been successfully implemented in several areas, such as cloud computing, scheduling, and pattern recognition [194].
The probabilistic and tortuous nature of lightning discharges originates from thunderstorms.Cloud-to-ground flashes are the most investigated occurrence in lightning research.During thunderstorms, a strong electric field is generated, which triggers an electron avalanche.In this case, it causes the negative coronal streamers and produces a current wave.As the current wave reaches the tip of the new leader, the coronal belt erupts and propagates outward, leading to the formation of a new space leader.This random growth process repeats continuously.Inspired by the phenomenon of lightning, the lightning search algorithm (LSA) [195] proposes 3 types of projectiles: transition projectiles for generating the initial leader population, space projectiles that attempt to become leaders, and lead projectiles representing the best solutions.A random distribution function based on the discharge probability characteristics and tortuosity characteristics of 3 projectile types is designed in LSA.LSA has been successfully applied to control systems, wind farm layout, and image segmentation [196].
In addition to the above methods, a series of paradigms based on physical principles have been proposed.Inspired by the attraction-repulsion mechanism between charged particles in an electromagnetic field, Birbil et al. proposed an electromagnetism-like mechanism algorithm [197], which has been applied to problems such as feature selection and flow shop scheduling [198].The lightning attachment procedure consists of 4 important stages: air breakdown on the cloud surface, downward movement of the lightning channel, upward leading propagation from the ground, and final jump.Inspired by this process, lightning attachment procedure optimization [199] has been successfully used to solve problems such as power generation scheduling [200] and intrusion detection [201].Inspired by Archimedes' principle, Hashim et al. [202] presented the Archimedes optimization.It mimics the principle of buoyancy exerted on an object that is partially or fully immersed in a fluid.The buoyancy is proportional to the weight of the displaced fluid.It has been used in industrial design [203].Transient search optimization [204], inspired by the transient behavior of switching circuits containing storage elements such as inductors and capacitors, has been applied in parameter estimation [205].

Geography
Rivers or streams are formed when water flows from one place to another, typically moving downhill until they reach the sea.Water evaporates from rivers, lakes, and plant leaves (through transpiration), forming clouds in the atmosphere.These clouds condense and release water back to Earth as rain or precipitation, completing the hydrological cycle (water cycle).Based on observations of the water cycle process, Eskandar et al. [206] proposed the water cycle algorithm (WCA), which has been widely used in combinatorial optimization [207] and landslide prediction [208].
Biogeography-based optimization (BBO) [209] is inspired by the principles of biogeography.Through the constant migration and drift of species between regions, nature finally reaches a state of equilibrium.The update mechanism for BBO relies mainly on 2 operations: migration and mutation.Migration is modeled as statistical models such as linear, cosine, quadratic, and exponential.These models describe the process of "the more the number of organisms in a place, the lower the inmigration rate and the higher the out-migration rate".The mutation process in BBO is analogous to that in GAs.BBO has been applied to problems such as feature selection [210], cloud computing [211], and shop floor scheduling [212].
In the atmosphere, winds flow to balance the air pressure.Specifically, it blows from a high-pressure region to a low-pressure region at a rate commensurate with the pressure gradient.When the air is in equilibrium and horizontal motion is stronger than vertical motion, the wind can be regarded as horizontal motion.The wind driven optimization (WDO) [213] is inspired by the flow of the atmosphere, where the movement of the wind can automatically compensate for the imbalance in atmospheric pressure.
According to Newton's second law, the motion law of a very small air unit is described.The final flow position of the air unit is used as a candidate individual to complete the modeling and solution of the problem.The velocities and positions of wind-controlled air masses in the WDO are renewed based on the physical equations governing atmospheric motion.WDO has been widely used in electromagnetics [214] and image segmentation [215].

Mathematics
The optimization process of a sine cosine algorithm (SCA) [216] is divided into 2 stages.In the exploration phase, the algorithm identifies a feasible region in the search space by combining random solutions from the set of candidate solutions.In the exploitation phase, these random solutions are gradually refined.The rate of change in the exploitation phase is lower than that in the exploration phase.In SCA, multiple initial random candidate solutions are generated, which fluctuate outward or in the direction of the optimal solution based on the mathematical model of sine and cosine.This approach enables the algorithm to explore different regions in the search space effectively.SCA has been successfully used in feature selection [217], parameter evaluation [218], energy scheduling [219], and image processing [220].
Gradient-based optimization (GBO) [221] is inspired by gradient-based methods, particularly Newton's method.The method mainly uses 2 kinds of operators: gradient search rule (GSR) and local escaping operator (LEO).A set of vectors is employed to explore the search space, with the GSR utilizing a gradient-based approach to accelerate convergence and achieve better positions within the search space.The LEO helps the algorithm escape local optima, enhancing its overall effectiveness.. Due to its simple structure, GBO has been widely used in various engineering optimization problems, such as parameter identification [222], model design [223], and human activity recognition [224]

Challenges and Potential Future Research Directions
Although the nature-inspired intelligent computing paradigm originated from diverse theories, it has developed its own unique trends over the decades.Nature-inspired intelligent computing paradigm presents innate advantages in solving real-world problems due to their parallelism, ease of expansion, and nonlinearity.The development of nature-inspired intelligent computing paradigms has moved forward in many research directions and differs from initial theories [13].By reviewing classical natureinspired intelligent computing paradigms, we summarize the top 10 potential future research directions and challenges, as shown in Fig. 14.

New extensions of the nature-inspired intelligent computing paradigm
Most of the existing nature-inspired intelligent computing paradigms primarily focus only on the evolutionary and phenological behaviors of organisms, often neglecting the growth and developmental processes of organisms, and fail to bridge the link between the macro and the micro.For example, while clustered regularly interspaced short palindromic repeats (CRISPR) genedriven evolutionary dynamics are noted in [225], similar models are largely absent in current evolutionary computations.
Combining different nature theories has become a popular topic.For example, through the integration of Lamarckianism and Darwinism [226], Lieberman et al. [227] combined evolutionary dynamics with graph representation.In addition, new rational mechanisms are constantly being explored and discovered to provide a source of new paradigms.Among group behaviors, Harpaz et al. [228] explain schooling behaviors in zebrafish, which converts visual input from neighbors into motor decisions.As scientific research progresses, more "empirical phenomena" will be explained, pointing toward new extensions of nature-inspired intelligent computing paradigms with theoretical implications.
However, it is crucial to recognize the potential risks of overreliance on "metaphors"."Metaphors" refers to paradigms in which there are unreasonable connections between natural mechanisms, mathematical models, and nature-inspired intelligent computing paradigms.Such weak or misleading connections can lead to the unnecessary reinvention of mechanisms and impede our understanding of these paradigms [30][31][32][33].Therefore, it is more important to clearly delineate the relationship between natural mechanisms and paradigms to substantially reduce the reliance on "metaphors".

Integration with the nervous system
Although different in function, the nervous and ISs work in specific ways to coordinate and regulate the function of the whole organism, working together to maintain the homeostasis of the internal environment.The variety of these systems does not operate independently of each other.For example, intestinal immune cells allow movement to reduce inflammation in the central nervous system [229], and the neuronal signals control the innate IS [230].Recently, Koren et al. [231] indicated that the brain stores and retrieves specific immune responses, extending the classical concept of immune memory to neuronal representations of inflammatory information.This finding underscores the inseparable connection between the brain and the IS, providing a stronger theoretical basis for immune computation.
Researchers have abstracted the neuronal structure of the nervous system into deep neural networks and built biological neural networks [232][233][234].A typical example is solving the temporal credit-assignment problem (TCA) by exploring the characteristics of spiking neurons.Gütig [232] developed an aggregate-label learning (AL) rule based on the responses of spiking neurons to the TCA problem.The derivatives of these responses indicate the most rapid changes in neuronal activity, enabling spiking neurons to match their output spikes with the number of clues or features present in the input data.Building on this, Qin et al. [233] introduced an innovative attentionbased loss function to solve the TCA problem.This function effectively integrates global temporal dynamics with intricate spike cluster data, thereby significantly enhancing the TCA capabilities of spiking neural networks (SNNs).In addition, several works combine evolutionary mechanisms with neural networks, such as the framework for evolutionary artificial general intelligence (FEAGI) [235], evolving probabilistic SNN (epSNN) [236], and EevoSpike NeuCube architecture [237].
In deep learning, backpropagation is commonly used to approximate.However, it is unlikely that the brain uses only backpropagation.Therefore, it is essential to "go back" to the biological sciences to uncover more meaningful evolutionary learning mechanisms to complement backpropagation mechanisms.Evolutionary RL is also a promising direction for developing deep neural networks that adapt to the complex environment [2,238,239].

Genotype-phenotype mapping
The genotype-phenotype mapping plays a crucial role in the design of EAs.This mapping is the process of mapping genes toward their biology functions [13].In algorithms, genotypephenotype mapping refers to the process of identifying the relationship between a system's hidden variable (genotype) and a measured observable quantity (phenotype) [240].Research [241] noted that the development of animal appearance phenotypes is controlled by large gene regulatory networks.Existing evolutionary-inspired paradigms lack expression of the complexity, flexibility, and plasticity of biological organs and ecosystems [13].
Traditional evolutionary paradigms typically employ a simple, direct genotype-to-phenotype mapping.However, the emergence of indirect coding leads evolutionary-inspired paradigms to deal with more complex systemic problems.As a result, how to better optimize the "genotype-expression" representation is also a topical and challenging issue for future evolutionaryinspired paradigms [242].
In recent years, large models have shown the potential to achieve general AI [243].These models' powerful representation capabilities present new opportunities for genotype-phenotype mapping.Traditional genotypes, often carefully designed based on expert knowledge, have limited generalization performance [13].Multimodal information such as natural language and vision can be directly used to describe problem characteristics.Directly searching for this information by vision and language models may provide a feasible way to achieve general optimization [84,244].Therefore, developing novel nature-inspired intelligent computing paradigms based on language and visual representations is a promising direction.

Large-scale problems
In reality, many physical world problems can be formulated as nondeterministic polynomial problems (NP-hard problems), such as transportation, industrial design, and data mining.Existing approaches to NP-hard problems are roughly categorized into 2 broad categories: exact algorithms and approximate algorithms.Exact algorithms, attempting to find the global optimal solution, consist of methods such as the branch-and-bound and the branchand-cut.The nature-inspired intelligent computing paradigm provides a way to approximate the solution of NP-hard problems [245].However, as the problem dimensionality increases, the problem complexity usually increases with the problem size, as well as the solution space of the problem grows exponentially with it.This results in a combinatorial explosion, and the performance of most available optimization algorithms degrades rapidly.
To further illustrate that large-scale problems are an important future research direction, we test the performance of popular biological-based paradigms in Table 1 on the benchmark problem Ackley at different problem scales.Figure 15A shows the average fitness value as a function of problem size.The x axis refers to the dimension of the decision variable, which varies from 5 to 200.It is obvious that the performance of all algorithms decreases as the problem size increases.To further reflect the performance changes of BWO, FA, and GWO, Fig. 15B shows the performance change curve at a larger problem scale (500 to 20,000).Experimental results still illustrate that these algorithms suffer performance collapse when dealing with large-scale problems.In the future, the applicability of the nature-inspired intelligent computing paradigm to large-scale problems should be further theoretically investigated.
Although nature-inspired intelligent computing paradigms are relatively efficient in obtaining the "best" solution through a search strategy rather than the exact approaches, they are still computationally expensive compared to traditional optimization methods such as Newton's method and hill climbing.Therefore, how to reduce costs through surrogate models, cooperative coevolution algorithms (CCPSO [246], MLCC [247]), and other theoretically guaranteed methods are also promising research topics for future research [19].

Multi-objective optimization
Multi-objective optimization (MOO) problems are broadly found in real-life applications such as recommendation systems, schedule cost problems, and scheduling decisions [248].In MOO, decision-makers need to optimize multiple tasks simultaneously.The trade-offs between different objectives result in a set of incomparable Pareto-optimal solutions that do not dominate each other.When confronted with MOO problems with more than 4 objectives, the majority of methods suffer from severe performance degradation.This is due to the increased computational cost of evaluating the objective function, as well as the rapidly increasing number of nondominated solutions breaking the Pareto selection pressure [249].Therefore, it is important to further explore the algorithm design [250], visualization [251], and theoretical analysis [252] of many-objective optimization problems.In addition, it is also challenging to solve large-scale MOO problems with irregular Pareto fronts [248].

Multi-task learning and optimization
Multi-task optimization (MTO) aims to solve multiple optimization tasks simultaneously by exploiting the commonalities and differences between tasks [253].In practical applications, related optimization tasks are ubiquitous.The populationbased paradigm has natural implicit parallelism, which is highly compatible with MTO.So MTO has gradually flourished in the field of evolutionary computing in recent years [254].However, with the increase in the number of optimization tasks, avoiding the impact of negative transfer remains an open challenge [255,256].
Multi-task learning (MTL) is considered to reflect the human learning process more accurately than single-task learning [257].MTL aims to solve multiple learning tasks simultaneously by exploiting the commonalities and differences between tasks.MTL expects to learn a machine to solve multiple learning tasks simultaneously, which can be regarded as a MOO problem.Establishing a unified MTL paradigm through MOO is an ongoing and significant area of research [258,259].

Parallel distributed computing
Nature-inspired intelligent computing paradigms have proven to be an effective method for solving NP-hard problems in practical applications due to their efficient parallelism and distributed nature.Given the rapid progress of the information age and the emergence of big data, NP-hard problems grow in scale and complexity.Existing methods are computationally expensive due to the large search space and expensive fitness evaluation [260].Furthermore, most paradigms can only be iterated serially in computers.Therefore, the parallel distributed nature-inspired intelligent computing paradigms are focused on by researchers.Many studies have explored parallel and distributed versions of nature-inspired intelligent computing paradigms, which fully exploit its parallelism to reduce computation time [261].The most intuitive idea is to directly use multiple distributed computing resources to perform parallel operations at the hardware level.In addition, distributed EAs are divided into 2 types at the software level: "distribution of population" and "distribution of dimensional" [249].The "distribution of population" model distributes individuals in a population across multiple processors or compute nodes, such as the master-slave approach and the island approach, while the "distribution of dimensional" model distributes problem dimensions, such as the agent-based approach and the coevolutionary approach.Furthermore, there is extensive research on integrating MOO problems with parallel distributed computing [262,263].Improving the scalability and convergence of parallel nature-inspired intelligent computing paradigms is currently the most difficult challenge.

Trusted federated learning
Safe and efficient federated learning is an important machine learning paradigm to achieve privacy protection, which can meet the needs of users and market regulation.Trusted federated learning is indispensable in any multi-party AI modeling process, with privacy protection, model performance, and algorithm efficiency at its core.Due to the parallelism and gradientfree, nature-inspired intelligent computing paradigms can enhance fair and trusted federated learning.These paradigms contribute to achieving a general AI paradigm that ensures supervised, controllable efficiency, and interpretable decisionmaking [264,265].

Automatic learning
Automated machine learning (AutoML) is an automation technology that enables machine learning models to adaptively solve real-world tasks [266].Generally, automatic deep learning (ADL) is composed of neural network architecture search (NAS), network parameter optimization, rule learning, and hyperparameter configuration.These problems present challenges such as nonlinearity, nonconvexity, high trial-and-error costs, and combinatorial explosion.
NAS [267] is one of the most popular research areas in AutoML.NAS based on RL is often limited by computational costs and can be unstable.Gradient-based methods often fall into local optima, leading to the discovery of ill-conditioned architectures [87].Due to their gradient-free, self-adjusting, and self-evolving capabilities, a series of nature-inspired intelligent computing paradigms have been proposed to solve the above problems in recent years [37,268,269].These techniques can flexibly provide low-cost solutions to complex NAS problems.With the development of general AI, large model-based evolutionary NAS provides a promising direction to further reduce the cost of architecture evaluation [270,271].
For network parameter optimization, gradient-based methods are commonly used.But they are subject to limitations such as "scaling problem" and easy to fall into a massive number of local optima and saddle points [272].EAs with stochasticity provide theoretical guidance for jumping out of saddle points [273].In rule learning, nature-inspired intelligent computing paradigms should be further explored in areas such as loss function search, self-evolution learning mechanism, and meta-learning [34].For the hyperparameter configuration, the impact of each hyperparameter on a deep learning model is interdependent [274].A set of hyperparameters is critical to the performance of the model, which relies heavily on the researcher's tuning experience and resources.Simple automated methods based on grid search and random search encounter the effects of dimensional catastrophe.The large-scale EA provides a promising solution for solving hyperparameter optimization [248].

Robotics based on biological-inspired paradigms
Biological-inspired robots aim to design robots that can dynamically interact with their environment.Such robots tend to help humans work in harsh conditions [275].A prominent example is the Spot robot dog, which mimics the behavioral movements of a dog [276].In addition, octopus robots, which are soft-bodied and highly agile, emulate the movements of octopuses [277].
In swarm robotics, multiple robots collaborate as a system to solve real-world problems by interacting with each other and combining their individual actions [278].Such robots are characterized by flexibility, scalability, robustness, autonomy, self-organization, self-assembly, and decentralization [279].Biologically inspired swarm robots leverage swarm behaviors observed in nature.Bredeche and Fontbonne [280] introduced social learningrelated algorithms for swarm robot deployment.Li et al. proposed a mobile robot based on an improved artificial fish swarm algorithm for path planning [281].In the future, it is also an intriguing direction to combine biological-inspired paradigms with biology to create swarm robots (artificial life) in natural systems.
Robots based on swarm intelligence mechanisms [282] also inspire new technology such as nanorobots in medicine [283], robots in space, and social robotics for specific groups [284].Beyond hardware limitations, there is still significant potential for research on the safety, reliability, and communication efficiency of swarm robots [285,286].

Summary and Outlook
Nature-inspired intelligent computing paradigms draw inspiration from the myriad of astounding rules and phenomena observed in nature.Over the years, these paradigms have offered potent solutions to a wide range of practical and complex challenges.This review provides a comprehensive summary of the natural mechanisms underpinning these advanced paradigms, categorizing them into 4 groups: evolutionary-based, biologicalbased, social-cultural-based, and science-based, each reflecting a unique aspect of the natural world.Additionally, the paper delves into the complex connections between these paradigms and natural mechanisms, revealing the commonalities of natureinspired intelligent computing paradigms.These commonalities provide a solid algorithmic foundation to avoid designing unreasonable "metaphors".
Based on a detailed analysis of natural mechanisms, we summarize key challenges facing the nature-inspired intelligent computing paradigm.The development of general AI offers new opportunities to address these challenges.Benefiting from flexible adaptability, nature-inspired intelligent computing paradigms can be combined with advanced large models, such as ChatGPT, for collaborative optimization.Inspired by lessons learned from nature, these synergistic mechanisms have the potential to foster more adaptive and efficient AI systems.

4 .
Challenges and future directions: The current challenges and future potential research directions based on the natureinspired intelligent computing paradigm are presented.This comprehensive discussion provides valuable future exploration for general artificial intelligence (AI) technology.

Fig. 6 .
Fig. 6.A general framework based on the biological-inspired paradigm.(A) Process of updating in the PSO algorithm, where P best and G best represent the local and global optimal positions of the particle swarm x i , respectively.(B) Main steps of the biological-inspired paradigm.

d
Lower and upper bounds for the dimension.LévyLévy flight function.
τ abHeuristic information (e.g., inverse of the distance for TSP) for edge (a, b).allowed kThe set of nodes that agent k can visit next.deposited on the path between nodes a and b by all m agents.Δ k abThe amount of pheromone deposited on the path by agent k.It is usually determined by the quality of the solution found by agent k.Δτ ab = Q/L k Amount ofpheromone deposited if agent k traveled edge (a, b), or 0 otherwise.Q is a constant; L is the length of the tour made by agent k. (t) i,d

Fig. 7 .
Fig. 7.The moth moves in a spiral at the same angle as the light.

Fig. 8 .
Fig. 8.The adaptive immune response of the immune system.

Fig. 9 .
Fig. 9. AIS usually considers the antigen as the problem to be solved and the antibodies as the potential solutions.
proposed a dendritic cell algorithm (DCA) based on the behavior of dendritic cells, which is widely used in intrusion detection problems.The algorithm generates a certain size population of dendritic cells and selects key attributes in the elements of the training set.These attributes are mapped to different types of signals, including security signals, danger signals, and pathogen-associated molecular pattern (PAMP) signals for solving intrusion detection problems.

Fig. 10 .
Fig. 10.A simple diagram of clonal selection theory.It shows the cloning, proliferation, and mutation of B lymphocytes in response to antigen stimulation, and the generation of plasma cells as well as memory cells.

Fig. 11 .
Fig. 11.In the ICA, the best countries (with lower costs) were picked as imperialist countries, while the rest as colonies.

Fig. 13 .
Fig.13.The performance of SA with 4 different strategies for optimizing the Boltzmann machine, including high T eff to low , low T eff to high T eff , low T eff , and high T eff (image from[287]).

Fig. 14 .
Fig. 14.Challenges of the nature-inspired intelligent computing paradigm and potential future research directions.

Fig. 15 .
Fig. 15.Trend of the average fitness value (smaller is better) with problem scale.

Table 1 .
Strengths and weaknesses of mainly social behavior-inspired paradigms

Table 2 .
Classical paradigms based on foraging behavior

Table 3 .
Common parameter introduction in Tables 2, 5, 7, 9, and 12 Position of agent i in dimension d at t iteration.The velocity of agent i in dimension d at t iteration.

Table 4 .
Therefore, researchers have developed various algorithms inspired by reproductive Parameter introduction corresponds to Table 2 Variable Content w The inertia weight, which controls the influence of the previous velocity on the new velocity.It helps in balancing exploration and exploitation.

Table 5 .
Classical paradigms based on navigation and migration behaviors i = d ij ⋅ e b⋅l⋅ cos 2 l + y j

Table 6 .
Parameter introduction corresponds to Table5

Table 7 .
Classical paradigms based on reproductive behavior

Table 8 .
Parameter introduction corresponds to Table7

Table 9 .
Classical paradigms based on other biological behaviors

Table 11 .
Classical paradigms based on AIS

Table 13 .
Parameter introduction corresponds to Table12

Table 12 .
Classical paradigms based on social and cultural patterns of behavior

Table 14 .
Classical paradigms based on science